67.5.17 problem Problem 3(f)

Internal problem ID [14032]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 6. Introduction to Systems of ODEs. Problems page 408
Problem number : Problem 3(f)
Date solved : Monday, March 31, 2025 at 08:22:30 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+x \left (t \right )-z \left (t \right )&=0\\ \frac {d}{d t}y \left (t \right )-y \left (t \right )+x \left (t \right )&=0\\ \frac {d}{d t}z \left (t \right )+x \left (t \right )+2 y \left (t \right )-3 z \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.119 (sec). Leaf size: 47

ode:=[diff(x(t),t)+x(t)-z(t) = 0, diff(y(t),t)-y(t)+x(t) = 0, diff(z(t),t)+x(t)+2*y(t)-3*z(t) = 0]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 +c_2 t +c_3 \,{\mathrm e}^{3 t} \\ y \left (t \right ) &= -\frac {c_3 \,{\mathrm e}^{3 t}}{2}+c_2 +c_1 +c_2 t \\ z \left (t \right ) &= c_2 +4 c_3 \,{\mathrm e}^{3 t}+c_1 +c_2 t \\ \end{align*}

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 132

ode={D[x[t],t]+x[t]-z[t]==0,D[y[t],t]-y[t]+x[t]==0,D[z[t],t]+x[t]+2*y[t]-3*z[t]==0}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{9} \left (-9 c_1 (t-1)+c_2 \left (6 t-2 e^{3 t}+2\right )+c_3 \left (3 t+2 e^{3 t}-2\right )\right ) \\ y(t)\to \frac {1}{9} \left (-9 c_1 t+c_2 \left (6 t+e^{3 t}+8\right )+c_3 \left (3 t-e^{3 t}+1\right )\right ) \\ z(t)\to \frac {1}{9} \left (-9 c_1 t-2 c_2 \left (-3 t+4 e^{3 t}-4\right )+c_3 \left (3 t+8 e^{3 t}+1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.130 (sec). Leaf size: 48

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(x(t) - z(t) + Derivative(x(t), t),0),Eq(x(t) - y(t) + Derivative(y(t), t),0),Eq(x(t) + 2*y(t) - 3*z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - C_{1} - C_{2} t + C_{2} + \frac {C_{3} e^{3 t}}{4}, \ y{\left (t \right )} = - C_{1} - C_{2} t - \frac {C_{3} e^{3 t}}{8}, \ z{\left (t \right )} = - C_{1} - C_{2} t + C_{3} e^{3 t}\right ] \]